### Why do this problem?

This
problem only requires some simple understanding of the
relationship between time, distance and speed, but it will require
clear thinking and insight to solve. It provides a good experience
of the mathematical 'lightbulb' moment when a seemingly tricky
problem is solved using a clever trick or alternative
viewpoint.

### Possible approach

Show the 'film' asking pupils to focus on the fly. Ask them
what data they would need to analyse the situation mathematically,
and provide the speeds and distance.

Ask students to begin calcuating times, distances and
positions of the trains and the fly. They will need to establish a
clear way of recording their results. Encourage them to describe
their strategy, even if they are frustrated by the length of the
task.

It is useful to discuss with students that in some
circumstances, the only option for solving a problem is to find a
long, complex algorithm. However, the frustration CAN be a trigger
to step back and find more efficient methods, and in this case
there is a good method.

The simple way of looking at this problem is the total time
until collision of the two trains and the complicated way is
working out the points at which the fly hits a train on each part
of the journey and then adding these up. Ideally someone in the
group will come up with the simpler method - if not, you could
leave it as an open problem, or begin to encourage the group to
focus on time.

### Key questions

- How long will it be until the fly first hits a train?
- How long will it be until the trains collide?

### Possible extension

Alter the distance between the engines or the speeds to change
the problem.

If the fly flew for 1 minute, what could the distance and
speeds have been?

If the fly flew 2000 metres, what could the distance and
speeds have been?

Speedy
Sidney provides a similar context for a slightly more
challenging problem.

This problem is a perfect opportunity to think about infinite
series - the fly makes infinitely many trips between the two trains
before collision. The total time taken for all of the trips must
equal the time taken for the trains to collide. Students might wish
to investigate the creation of this series by working out how long
the fly takes to travel on each portion of its journey.

### Possible support

As a warm up, you might try the simpler problem

Bike
Ride